Tóm tắt Luận án Estimation and computational simulation for the effective elastic moduli of multicomponent materials

VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY ------o0o------ VU LAM DONG ESTIMATION AND COMPUTATIONAL SIMULATION FOR THE EFFECTIVE ELASTIC MODULI OF MULTICOMPONENT MATERIALS Major: Engineering Mechanics Code: 62 52 01 01 SUMMARY OF PhD THESIS Hanoi – 2016 The thesis has been completed at: Vietnam Academy of Science and Technology Graduate University of Science and Technology Supervisors: Assoc. Prof. Dr.Sc. Pham Duc Chinh Reviewer 1: Reviewer 2: Reviewer 3: Thesis is defended at: on , Date Month Year 2016 Hardcopy of the thesis can be found at: 1 INTRODUCTION Homogenization for composite material properties has made great progress in scientific research. The construction of the material models was made very early and from the basic ones. The macroscopic properties of materials depend on many factors such as the nature of the material components, volume ratio of components, contact between elements, geometrical characteristics ... Therefore, the thesis is done with the purpose of building evaluations for macroscopic elastic moduli of isotropic multicomponent materials which yield results better than the previous ones. Topicality and significance of the thesis Multicomponent materials (also known as composite materials) are used widely life today. We may see composite materials that will be the key ones in the future because of flexibility and multipurpose material, however the identification of macroscopic materials is not easy because we often have only limited information about the structure of the composites. The objective of the thesis Construction of bounds on the elastic moduli of isotropic multicomponent materials which involve three-point correlation parameters. We use numerical methods to study several representative material models. 2 Study method • Using the variational approach based on minimum energy principles to construct upper and lower bounds for the effective elastic moduli of isotropic multicomponent materials. • The numerical method using MATLAB program to set the formula, matrix ... optimal geometric parameters for particular composite materials. CAST3M program (established by finite element method) has been applied to several periodic material models for comparisons with the bounds. New findings of the dissertation • Construction of three-point correlation bounds on the effective elastic bulk modulus of composite materials and applications of the bounds are given to some composites such as symmetric cell, multi-coated sphere, random and periodic materials. • Construction of three-point correlation bounds on the effective elastic shear modulus of composite materials and applications of the bounds are given to some composites. • FEM is applied to some compact composite periodic multicomponent materials for comparisons with the bounds. 3 Structure of the thesis The content of the thesis includes an introduction, four chapters and general conclusions, namely: Chapter 1: An overview of homogeneous material Chapter 1 presents a literature review of related work on homogeneous material previously obtained by the domestic and abroad researchers. Two methods to find effective elastic moduli, which are direct equation solving and variational approach based on energy principles, are briefly reviewed. Chapter 2: Construction of third-order bounds on the effective bulk modulus of isotropic multicomponent materials The author uses new a trial field that is more general than Hashin- Shtrikman polarization ones to derive three-point bounds on the effective elastic bulk modulus tighter than the previous ones and to construct upper and lower bounds of effk . Applications of the bounds are given to composite structures. Chapter 3: Construction of third-order bounds on the effective shear modulus of isotropic multicomponent materials The author constructs upper and lower bounds of effμ from minimum energy and minimum complementary energy principles. Applications of the bounds are given to some composites. Chapter 4: FEM applies for homogeneous material Calculations by the constructed FEM program for a number of problems with periodic boundary conditions are compared with the results from the two previous chapters. General conclusions: present the main results of the thesis and discuss further study. 4 CHAPTER 1. AN OVERVIEW OF HOMOGENEOUS MATERIAL 1.1. Properties of isotropic multicomponent materials Representative Volume Element (RVE) of multicomponent materials is given by Buryachenko [11], Hill [30]; RVE is “entirely typical of the whole mixture on average”, and “contains a sufficient number of inclusions for the apparent properties to be independent of the surface values of traction and displacement, so long as these values are macroscopically uniform”. Figure 1.1 Representative Volume Element (RVE) Consider a RVE of a statistically isotropic multicomponent material that occupies spherical region V of Euclidean space, generally, in d dimensions (d = 2, 3). The centre of RVE is also the origin of the Cartesian system of coordinates {x}. The RVE consists of N components occupying regions V Vα ⊂ of volume 1 , . .( . ,v Nα α = ; the volume of V is assumed to be the unity). The stress field satisfies equilibrium equation in V: V∇⋅ = ∈xσ 0 , (1.1) The local elastic tensor C(x) relates the local stress and strain tensor fields: ( ) : ( )=x C(x) xσ ε (1.2) 5 The effective elastic moduli ( , )kα αμ=C(x) T , where T is the isotropic fourth-rank tenser with components: 2( , ) ( )ijkl ij kl ik jl il jk ij klT k k d = + + −μ δ δ μ δ δ δ δ δ δ , (1.3) ijδ is Kronecker symbol. The strain field ( )ε x is expressible via the displacement field ( )u x 1 2 ( ) ( ) ;T V⎡ ⎤= ∇ + ∇ ∈⎣ ⎦x u u xε (1.4) The average value of the stress and strain has form: 1 1,= =∫ ∫σ ε V V d V d V σ εx x (1.5) The relationship between the average value of the stress and strain on V is given by effective elastic moduli effC : : , ( , ).= = efeff eff ffe fk μC C Tσ ε     (1.6) This is called the direct solving of the equation. In addition, a different approach to determine the macroscopic elastic moduli may be defined via the minimum energy principle (where the kinematic field ε is compatible): 0 0 0: : inf : :eff V d 〈 〉= = ∫ε εε ε ε εC C x (1.7) or via the minimum complementary energy principle (where the static field σ is equilibrated): 0 0 1 0 1: ( ) : inf : ( ) :eff V d− − 〈 〉= = ∫ xσ σσ σ σ σC C (1.8)  The variational approach can give the exact results but it will be upper and lower bounds, this is a possible result when we apply to the specific material that we do not have all information of material geometry. 6 1.2. An overview of homogeneous material From the late 19th century to early 20th century, the study of the nature of the ongoing environment of multi-phase materials received great attention from the leading scientists in the world. In the case of the model is two-phase materials with inclusion particles as spherical shape beautiful, oval (ellipsoid) distribution platform apart in consecutive phases (phase aggregate ratio is small), Eshelby [20] took out an inclusion of infinite domain of the background phase, and precisely calculated the stress and strain. On that basis, he found effective elastic moduli in a volume ratio vI region (inclusions apart each other). For models with the component materials distributed chaotically (indeterminate phase), it is difficult for pathway directly solving equations. Therefore, several methods are proposed. A typical model is differential diagram method (differentials scheme) in which stresses and strains calculated in step with the background of the previous step phase contain a small percentage of spherical inclusions or oval (using results of Eshelby). Finally, effective moluli of the mixture for the steps can be calculated. Besides getting answers through solving equations, there is another method to find out macroscopic properties of composite materials based on finding extreme points of energy function. Although not getting the stress field and strain field accurately corresponding to the extreme point, we still receive the corresponding bounds for extreme values of energy functions and the macroscopic properties of the material which is relatively close to the true value. 7 Hashin and Shtrikman (HS) [28] have built variational principle by using the possible polarization (polarization fields) with average values various across different phases. Their results for isotropic composite materials were much better than those of Hill-Paul. Of domestic studies, Pham Duc Chinh’s works considered the problem for the multi-phase materials when considering the difference of phase volume ratio, micro-geometries of the components that are characterized by three-point correlation parameters. In some cases, he found the optimal results (achieved by a number of specific geometric models). For the evaluation narrower than the rated HS, the following authors have researched and built the variational inequalities containing random function describing additional information about the geometry of the particular material phase. The random function of degree n (n - point correlation functions) depending on the probability of any n points is taken incidentally (with certain distance) and points fall into the same phase between them. Not from the principle of HS, but from the minimum energy principle and the HS polarization trial fields, Pham found a narrower HS’s estimations though part that contains information about geometry of materials. Another study on the homogeneous materials using numerical method with classic digital technique has built approximately from kinetic field possible. But there are also obstacles where it is difficult to find the simplest possible field over the entire survey area. In case the field is found, the system of equations may be large and complex to solve. These problems have been overcome by the fact that the local approximation, on a small portion of the survey area, has explanation and simultaneously and leads to neat equations and 8 calculations extent consistent with the possibility the system features high-speed computers. Approximation techniques smart elements (element-wise) have been recognized for at least 60 years ago by Courant [17]. There have been many approximation methods for solving elastic equations. The most popular is probably finite element method (FEM). The significance of this approach is the partition object into a set of discrete sub-domains called elements. This process is designed to keep the results of algebraic computation and memory management efficiency as possible. 9 CHAPTER 2. CONSTRUCTION OF THIRD-ORDER BOUNDS ON THE EFFECTIVE BULK MODULUS OF ISOTROPIC MULTICOMPONENT MATERIALS Three-point correlation parameters have been constructed and used by many authors in the evaluation and approximation of effective elastic composite materials. By choosing more general multi-free parameter trial fields than the ones of Hashin – Shtrikman, we constructed tighter three-point correlation bounds. 2.1. Construction of upper bound on the effective bulk modulus of isotropic multicomponent materials via minimum energy principle To construct the effective bulk modulus effk from (1.7), we choice the trial field as form: 0 1 1, ; , , , N ij ij ij a i j d d α α α= ⎛ ⎞ε = + ϕ ε = ⎜ ⎟δ⎝ ⎠∑ (2.1)  Where 0 0= ijij d δε ε is a constant volumetric strain, αϕ   is hamornic potential; aα are free scalar that satisfy the restrictions [for the trial field to satisfy the restriction 0ε = ε ], Latin indices after comma designate differentiation with respective Cartesian coordinates. Substituting the trial field (2.1) into energy functional (1.7), one obtains: 2 0 2 1 1 C x x 2 2 , , : ( ) : ( ) ( ) N N V V W d k v k a a A a aβγα α α α α α β γ α= α β γ= = = + + + μ⎡ ⎤⎢⎢ ⎥⎣ ε⎥ ⎦∑ ∑∫ε ε ε (2.2) Where: - 1= =∑NVk kvα α α  is called Voigt arithmetic average. 10 - xij ij V A d α βγ βα γα α = ϕ ϕ∫ is three-point correlation parameters. We minimize (2.2) over the free variables aα have restriction with the help of Lagrange multiplier λ and get the equations: 0 2 1 0 2 1 C x v v: : ( ) ' · · ( ) N V V k k k V W d k v k a k −ε α α α α= ⎡ ⎤ ⎡ ⎤= = + ε = − ε⎢ ⎥ ⎣ ⎦⎣ ⎦∑∫ε ε A (2.3) Where: - { } { }1 1 1 1v v' , , ( ), , ( );T Tk N N k R N N Rv k v k v k k v k k= = − −" " { } 1 1 1 1 2 , , , ,kk N N k R A N A v k A v k k A αβ αβ − δβ αβ α α αβ γ α δ γ γ γ= δ= = α β = ⎛ ⎞= δ + − μ⎜ ⎟⎝ ⎠∑ ∑ "A Taking in account (1.7) with (2.3), one obtains the upper bound on effk : 1v v({ , , },{ }) · ·eff UA V k k kk K k v A k βγ ′ − α α α α≤ μ = − A (2.4) 2.2. Construction of lower bound on the effective bulk modulus of isotropic multicomponent materials via minimum complementary energy principle To find the best possible lower bound on effk from (1.8) we take the following equilibrated stress trial field: 0 1 1,( ) ; , , , N ij ij ij ija I i j d α α α α= ⎡ ⎤σ = δ + ϕ −δ σ =⎢ ⎥⎣ ⎦∑ " ; (2.5) with I α is an indicator function. Substituting the trial field (2.5) into (1.8) and following procedure similar to that form, one obtains: 11 0 2 1 0 2 1 1 v v( ) ' · · ( ) N kkR R k a vd W k k d k −− −α α σ αα= ⎡ ⎤− ⎡ ⎤= − σ = − σ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦∑ A (2.6) 11 Where: - 1 1 N R v k k − α αα= =∑ is called Reuss harmonic average , - 1 1 1 11 1 1 1v ( ), , ( ) T k V N N V d d v k k v k k d d − − − −− −⎧ ⎫= − −⎨ ⎬⎩ ⎭" , - { } 2 1 1 2 1 1 1 2( ) ( ,) k k N N k V A vd A v k A k A kd αβ − αβ δβ −ααβ α α αβ γ δ γ γ γ= δ= = ⎛ ⎞−= δ + − μ⎜ ⎟⎝ ⎠∑ ∑ A - 1 11 1 1 1v ' , , T k N N d d v k v k d d − −− −⎧ ⎫= ⎨ ⎬⎩ ⎭" . The best possible lower bound on effk has been identified: 11 1v v({ , , },{ }) ( · · )eff L k kkA Rk K k v A k ′ −βγ − − α α α α≥ μ = − A (2.7) 2.3. Applications 2.3.1. Two-phase coated spheres model (a) (b) 12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 2 4 6 8 10 12 14 16 18 v2 ke ff HS DXC 3D (c) Figure 2.1 Bounds on the elastic bulk modulus of two-phase coated spheres and symmetric spherical cell mixture at 1 1 2 2 21, 0.3, 20, 10, 0.1 0.9= = = = = →μ μk k v . (a) Coated spheres; (b) Symmetric spherical cell mixture; (c) HS - Hashin-Strikman upper and lower bounds and also the respective exact effective bulk moduli of the coated spheres at 2 1=ζ và 1 0=ζ , DXC 3D - upper and lower bounds for the symmetrical spherical cell mixtures. 2.3.2. Two-phase random suspensions of equisized spheres Now consider the two-phase random suspensions of equisized hard spheres (Fig. 2.5a) and overlapping spheres (Fig. 2.6a) in a base phase-1. The bounds (2.4) and (2.7) for the models at 2 1 1 2 20.1 0.99, 1, 0.3, 20, 10,= → = = = =kv kμ μ together with Hashin-Shtrikman bounds are projected in Fig.2.2b, Fig.2.3b. 13 (a) 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 1 2 3 4 5 6 7 8 9 v2 ke ff HS KCL 3D (b) Figure 2.2 Hashin-Strikman bounds (HS) and the bounds (KCL 3D) on the elastic bulk modulus of the random suspension of equisized hard spheres (a) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 20 v2 ke ff HS CL 3D (b) Figure 2.3 Hashin-Strikman bounds (HS) and the bounds (CL 3D) on the elastic bulk modulus of the random suspension of equisized overlaping spheres 14 Comment: In Figure 2.2b shows the lower bound that approaches bound HS and the upper bound also quite far apart because kα differences between inclusion (spheres) and the matrix. In figure 2.3b, lower bound still tend to approach the lower bound of HS but upper bound also closes to the upper HS bound at 2 0 99.v = . 2.3.3. Three-phase doubly-coated sphere model We come to the three-phase doubly-coated sphere model (Fig. 2.4a), where the composite spheres of all possible sizes but with the same volume proportions of phases fill all the material space - an extension of Hashin-Shtrikman two-phase model, at the range 1 1 2 2 3 312 8 1 0 3 30 15, , , . , , .k k k= μ = = μ = = μ = (a) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2 3 4 5 6 7 8 9 10 11 12 v1 ke ff HS PDC 1996 NCX 3D (b) Hình 2.4 Bounds on the elastic bulk modulus of doubly coated spheres at the range 1 2 3 1 10 1 0 9 1 2 . . , ( )v v v v= → = = − . (a) Doubly coated spheres; (b) Hashin-Strikman (HS), old bounds (PDC 1996) and the new bounds (NCX 3D) which converges to the exact value of the effective bulk modulus 15 Comment: Figure 2.4b shows the results of three-phase doubly coated spheres model. In the case of two-phase coated spheres model, the PDC 1996 bounds are convergence, but it is not convergence in the case of three-phase, also three-point correlation parameters of the material has considered. The results are the same between the upper and lower bounds, a new contribution of the thesis. 2.3.4. Symmetric cell material model Lastly we come to the symmetric cell material (Fig. 2.5a) without distinct inclusion and matrix phase (Pham [50], Torquato [77]). effk fall inside Hashin-Shtrikman bounds for the large class of isotropic composites. (a) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 2 4 6 8 10 12 14 16 v1 ke ff HS TDX 3D (b) Figure 2.5 Hashin-Strikman bounds (HS) and bounds (SYM) on the elastic bulk modulus of three-phase symmetric cell mixtures (TDX3D), 1 2 3 10 1 0 9 0 5 1. . , . ( )v v v v= → = = − 1 1 2 2 3 31 0 3 12 8 30 15, . , , , ,k k k= μ = = μ = = μ = (a) A symmetric cell mixture; (b) The bounds 16 2.4. Conclusions On the variational way the author has constructed the upper and lower bounds effk of isotropic effective elastic material through the minimum energy and minimum complementary energy principles. Lagrange multiplier method is used to optimize the energy function with free variables aα have restriction. It was found that the trial fields which are chosen (contain N - 1 free parameters) more general than those in [1] contain only one free parameter, as compared in detail in the case of three-phase doubly- coated sphere model. Some models built in case of d-dimensional space so the results are used in the general case and the bounds contain the properties, volume fractions of the component materials and the three-point correlation parameters that contain information about the geometry of the material phase to give the better results. The results were applied to some specific material models such as two-phase coated spheres model, two-phase random suspensions of equisized hard spheres, three-phase doubly-coated sphere, symmetric cell material in space 2D and 3D. To be clear, in the calculation of comparison, the author chose the material properties varying widely. The small difference of estimations comes closer to each other for an approximate value of macroscopic material properties. Results in this chapter are published by the author in the scientific works 1, 2, 4 and 5. 17 CHAPTER 3. CONSTRUCTION OF THIRD-ORDER BOUNDS ON THE EFFECTIVE SHEAR MODULUS OF ISOTROPIC MULTICOMPONENT MATERIALS Similar to chapter 2, the method is also based on energy principle to help identify the upper and lower bounds of effective shear modulus of isotropic multicomponent materials. 3.1. Construction of upper bound on the effective shear modulus of isotropic multicomponent materials via minimum energy principle To construct the effective shear modulus effμ , we choice the trial field as form: 0 0 0 0 1 1 1 2 , , , ( ) , , ,... ;, N ij ij ik kj jk ki ijkl kla b i j d α α α α α α= ⎡ ⎤ε = ε + ϕ ε +ϕ ε + ψ ε =⎢ ⎥⎣ ⎦∑    (3.1) Where 0 0 0 0( )ij ij iiε = ε ε =  is a constant deviatoric strain, αψ is biharmonic potential, ,a bα α are free variables that have restricted. Substituting (3.1) into (1.7), one obtains: ( )1 0 0C x v v 2 : : · · .V ij ij V W d −ε μ μ μ′= = μ − ε ε∫  ε ε A (3.2) From (1.7) and (3.2), finally we obtain the upper bound on the effective shear modulus effμ : 1v v({ , , },{ , }) · ·eff UAB VM k v A B βγ βγ ′ − α α α α α μ μ μμ ≤ μ = μ − A (3.3) We have introduced vectors v v,′μ μ and matrix μA in 2N-space: { } 1 1 1 1 1 1 1 1 1 2 2v 2 2 1 2 2 2v 2 2 ( ) ( ) ( ), , ( ), , , , ( ) ( ) , , , , , , , , , , ( ) , ( ) . T N R N N R R N R T N N N N N V v v v v d d d d d d A N v v v v d d d d d v d μ μ α α α μ α = β μ μ −μ μ −μ ⎫⎧= μ −μ μ −μ⎨ ⎬+ +⎩ ⎭ = α β = μ μ μ μ μ⎧ ⎫′ == ⎨ ⎬+ +⎩ ⎭ μ∑ " " " " " A 18 3.2. Construction of lower bound on the effective shear modulus of isotropic multicomponent materials via minimum complementary energy principle To find a lower bound on the effective shear modulus effμ , we take the admissible equilibrated stress trial field: ( )0 0 0 0 0 0 1 1, , , ,( ) , , ,.. , ;. N ij ij ik kj jk ki ij ij kl kl ijkl kla I a b b i j d α α α α α α α α α α= ⎡ ⎤σ = σ + ϕ σ +ϕ σ − σ − + δ ϕ σ + ψ σ =⎦⎣∑      (3.4) Where 0 0 0 0( )ij ij iiσ = σ σ =  is a constant deviatoric stress. Substituting the trial field (3.4) into (1.8) and following procedure similar, one obtains the best possible lower bound on effμ : ( ) 11 1v v{ , , },{ , } ( · · )eff LAB RM k v A B ′ −βγ βγ − −μ μμα α α α αμ ≥ μ = μ − A (3.5) Where: { } 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 2 22 2 v 2 2 2 2 2 2 v 2 2 1 2 ( ) ( )( ) ( ) ( ), , ( ), , , ( ) ( ) ( ) ( ) , , , , , ( ) ( ) , , , , , ; T V N N VN V N V N N N N R v vd v d v d d d d d d d v d v v v d d d d d d v A N − − − − − − − − μ − − − −′ μ μ − ααβμ αα ⎧ ⎫μ −μ μ −μ− −⎪ ⎪= μ −μ μ −μ⎨ ⎬+ +⎪ ⎪⎩ ⎭ ⎧ ⎫− μ − μ μ μ⎪ ⎪= ⎨ ⎬ μ + +⎪ ⎪⎩ ⎭ = α β = = μ " " " " "A 1 . N = ∑ 3.3. Applications 3.3.1. Symmetric cell material model This material model without distinct inclusion and matrix phase Pham [50] in 3D-space (Fig.3.1a), the three-point correlation parameters ,A Bβγ βγα α have particular forms [50-51]. 19 (a) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 1 2 3 4 5 6 7 8 9 10 v1 μef f HS TDX 3D DXC 3D (b) Figure 3.1 Bounds on the effective shear modulus of three- component symmetric cell materials (TDX 3D), compared to bounds for the specific symmetric spherical cell materials (DXC 3D) and Hashin-Shtrikman (HS) bounds; 1 2 3 10 1 0 9 0 5 1. . , . ( )v v v v= → = = − with 1 1 2 2 3 31 0 3 12 8 30 15, . , , , ,k k k= μ = = μ = = μ = . (a) A symmetric cell mixture; (b) Bounds Comment: The thesis’s estimations are tighter than those before (HS bounds), and in 3D- space in range 1 0 1 0 4. .v = → , the upper bound UDXCμ and upper bound UTDXμ are the same. According to the opposite direction in range 1 0 5 0 9. .v = → the lower bound LDXCμ and lower bound LTDXμ are also the same. 3.3.2. Periodic two-phase model with hexagonal in shape Lastly we come to periodic two-phase model with hexagonal in shape (LGD) (Fig. 3.2a) in range 2 0 1 0 7. .v = → with 1 1 2 21 0 5 10 5, . , ,k k= μ = = μ = . Two parameters 1ζ (or 2ζ ) and 1η (or 2η ) for this material are given in [77]. 20 (a) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.5 1 1.5 2 2.5 3 v2 μef f HS LGD (b) Figure 3.2 Bounds on the effective shear modulus of periodic hexagonal material (LGD) and Hashin-Shtrikman (HS) bounds The results show that LGD bounds are tighter than HS bounds. 3.4. Conclusions The author has developed an unified approach to construct three- point correlation bounds on the effective conductivity and elastic moduli of statistically isotropic N-component materials from minimum energy principles, using multi-free-parameter trial fields, which are generalizations of Hashin–Shtrikman two-free-parameter polarizationfields [1], [49]. The trial fields include 2 2N − free parameters compared with 2 parameters ( 0 0,k μ ) of [1], [49], to construct new tighter bounds at 3N ≥ . Bounds are specified to the practical class of symmetric cell materials and two-phase periodic material. Results in this chapter are published by the author in the scientific works 1, 3 and 5. 21 CHAPTER 4. APPLICATION OF FINITE ELEMENT METHOD TO PERIODIC MULTICOMPONENT MATERIALS This chapter describes homogeneous theory for periodic materials with and the assumptions for the application of the FEM which runs on the open source code of the program CAST3M (France). Results are calculated for a specific periodic model and they are compared with the estimations in chapter 2 and chapter 3. 4.1. Homogenization of periodic materials The idea of this theory is the basic information that is related to the physical properties of the microstructure and stored in a base structure (periodic cell). Then, a periodic pattern for the actual material can be achieved by filling the entire space with the base structure in a cyclic. Figure 4.1 Basic structure of the periodic material Based on the theory of homogenization, the calculations make on specific element. To solve the problem we considered characteristic elements of the research material symbols Ω affected by a strain homogeneous fieldE . This strain field is produced by an average stress field across the region Σ . 22 4.2. Introduction to CAST3M program CAST3M program is supported by technology research organization under the French government with a history of 20 years. This program contains the necessary elements to simulate the object calculated by FEM. Scope of application includes mechanical behavior of elastic materials, elastic - visco - plastic... As well as the structure of a program computed by the finite element method, CAST3M is a open source code for the analysis of structures that also includes the steps of: • Create the respective finite element meshes. • Provide the respective material and mechanical properties. • Establish the periodic boundary conditions • Impose a macroscopic uniform stress • Calculate the macroscopic elastic properties 4.3. Application for specific material In the case of transverse-isotropic unidirectional composites is studied, RVE is shown in Fig 4.2 Figure 4.2 Unit cell of periodic material 23 The fiber reinforced cylindrical shaped shaft runs vertical axis and the cross section is arranged in hexagonal shape. Five effective elastic moduli of this material are specified as follows: effective area modulus 1 2( , ) effe e K , effective shear modulus 1 2( , ) effe e μ ,  Poisson's ratio in 1 3( , )e e  or 2 3( , ) effe e ν , axial elastic modulus effE and axial shear modulus in 1 3( , )e e  or  2 3( , ) effe e μ .  The transverse-isotropic unidirectional composites have matrix phase and inclusion phase denoted m and i, respectively. Two phases are linear elastic and isotropic, and characterized by E andν. The value of 5 effective elastic moduli depends on the percentage of the volume fraction of inclusion phase iv that will be presented from figure 4.3 to figure 4.7. The Fig. (a) corresponds to the data: 1 0 25 10 0 35, . , , .m m i iE E= ν = = ν = and Fig.  (b) corresponds to the data: 10 0 35 1 0 25, . , , .m m i iE E= ν = = ν = . FEM results are indicated by bold dashed line and two solid lines performance the upper and lower bounds. Results for effK , effμ are taken by chapter 2 and 3 for 2D-space problem. Estimations of effE and effν rely on Hill’s relations for two- phase transverse-isotropic unidirectional composites. Estimation of effμ is the same with effective conductivity of two-phase isotropic composite that uses results in [45]. 24 Figure 4.3 Relationship eff iK v− Figure 4.4 Relationship eff ivμ − Figure 4.5 Relationship eff ivν − 25 Figure 4.6 Relationship eff iE v− Figure 4.7 Relationship eff ivμ − Comment: From figure 4.3 to 4.7, numerical results (bold dashed line) line between the evaluations (solid line), especially effE and effν are coincided (Figure 4.5, 4.6). The results of effE and effν are determined after we obtain effK . It is noted that upper bound of effK may not give upper bound of effE , effν but may give lower bounds of effK (surveying function is monotonous!). 26 4.4. Conclusions The author has presented theoretical homogenization for periodic composite material within the framework of the thesis. It can be the basis for further studies. The author has calculated several effective elastic moduli by FEM for transverse-isotropic unidirectional composites with hexagonal cross section. The problems of periodic composite material on the boundary force and displacement conditions have differences compared with the common FEM calculations. Here, the material is periodic so problem should be taken on periodic cell with boundary condition that is periodic displacement while condition of forces is anti-periodic. The results were displayed and compared with the ones calculated by estimations which built in the previous chapters. The results of FEM line between the evaluations (which are almost coincident) confirming the reliability of the method. The drawback of the method is calculated primarily for periodic materials. For materials with randomly arranged forms such as random spheres mentioned in the previous chapter, it will be difficult for calculation because we must consider for RVE with size rather than unit cell. The results presented in this chapter are published in references 5 and 6. 27 GENERAL CONCLUSIONS The thesis has constructed estimations for effective elastic moduli of isotropic multicomponent materials, which involve three-point correlation parameters. FEM is applied to a periodic composite material for comparisons with the bounds. New findings of the dissertation • Construction of new estimations for effective elastic moduli of isotropic N-component materials in d- dimensional space. The thesis has used the general polarization fields contain more free variables than the fields in [1], to get the evaluation simpler and better in the 3N ≥ . Lagrange multiplier method is applied to optimize the energy functional containing free variables that are subjected to the constraints. The new estimations contain information on the properties, volume proportions of the component materials and three-point correlation parameters about the geometry of the material. • The new estimations are applied to some models with known three-point correlation parameters: symmetric cell material model, two-phase random suspensions of equisized spheres, multi-phase coated spheres model (a rather interesting finding is when applied to three-phase doubly-coated sphere model, the results for the upper and lower bound are coincident, to be the exact moduli), periodic models in 2D-space and 3D-space. 28 • Using MATLAB program to set the vector, matrix in calculating the new estimations to optimize geometric parameters for specific materials. CAST3M program is used to calculate the moduli of two-phase transverse- isotropic unidirectional composites (cross section is hexagonal symmetry) for comparisons with the evaluations. The thesis constructed the numerical program which can help to designed and predict new material properties, as desired. Issues for further research • Construct the trial fields for better evaluation (narrower upper and lower bound). • Build further evaluations for more sophisticated materials such as random cell polycrystals. • Combine estimations with the FEM simulation and approximation methods to study the properties of complex geometry materials. List of the Author’s publications 1. Pham, D.C., Vu, L.D., Nguyen, V.L. (2013), Bounds on the ranges of the conductive and elastic properties of randomly inhomogeneous materials. Philosophical Magazine 93, 2229-2249. 2. Pham Duc Chinh and Vu Lam Dong (2012), Three-point correlation bounds on the effective bulk modulus of isotropic multicomponent materials. Vietnam Journal of Mechanics 34, pp. 67-77. 3. Vu Lam Dong and Pham Duc Chinh (2013), Construction of bounds on the effective shear modulus of isotropic multicomponent materials. Vietnam Journal of Mechanics 35, 275-283. 4. Vũ Lâm Đông, Phạm Đức Chính (2012). Đánh giá bậc 3 mô đun đàn hồi diện tích của vật liệu đẳng hướng hai chiều nhiều thành phần. Hội nghị Cơ học toàn quốc lần thứ IX Hà Nội, 8-9/12/2012, 303-312. 5. Vũ Lâm Đông, Phạm Đức Chính và Trần Bảo Việt (2013). Đánh giá biến phân và tính toán số PTHH cho các hệ số đàn hồi vật liệu tổ hợp đẳng hướng ngang. Hội nghị Khoa học toàn quốc Cơ học Vật rắn biến dạng lần thứ XI Thành phố Hồ Chí Minh, 7-9/11/2013. 6. Trần Bảo Việt, Vũ Lâm Đông và Phạm Đức Chính (2014). Mô phỏng số PTHH và đánh giá các hệ số đàn hồi vật liệu cốt sợi dọc trục đẳng hướng ngang. Tuyển tập công trình Hội nghị Cơ học Kỹ thuật toàn quốc Kỷ niệm 35 năm thành lập Viện Cơ học, 10/4/1979-10/4/2014,Tập 2, 443-448.